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New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. (English) Zbl 0947.42024
Given a window function $\psi \in L^2(\Bbb R)$, the Gabor transform of a function $f\in L^2(\Bbb R)$ is given by $Gf(\omega,t)= \frac{1}{\sqrt{2\pi}} \int_\infty^\infty f(x) \overline{\psi(x-t)}e^{-i\omega x} dx$. It is proved that if $f\neq 0$, then the support of $Gf$ has infinite Lebesgue measure. A similar result is stated for the wavelet transform. An abstract framework including both cases is presented.

42C40Wavelets and other special systems
43A32Other transforms and operators of Fourier type
46E22Hilbert spaces with reproducing kernels
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